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SUMMARY: The foramen magnum (FM) is a transition zone between the spine and skull. There is a study in the literature showing the golden ratio (4.62) and harmony between both FM and the anteroposterior and transverse diameters of the skull. The aim of this study is to examine the existence of this ratio on CT images and to investigate whether this ratio changes according to the FM shape types. In this study, 402 adult CT images belonging to the Turkish population were examined. Maximum cranial length (MCL), maximum cranial width (MCW) and the FM length (FML) and FM width (FMW) were measured. The different shapes of the FM were macroscopically classified. The number and incidence of each type in the studied skull was registered. In the 402 CT images used in the study, 12 FM shapes were detected. Three of the shape types observed in our study have not been reported in the literature until now, and were found in our study for the first time. In addition, our data supported that by using the ratio of 4.62, there was a great harmony between the skull and FM. The results obtained from the research show that there is a ratio of 4.62 between the basic cranial measurements and FM dimensions in all shapes except triangular shape. It is thought that repeating the calculated coefficients over more different FM shapes will contribute to the effectiveness of the proposed golden ratio.

RESUMEN: El foramen magno (FM) es una zona de transición entre la columna y el cráneo. Existe un estudio en la literatura que muestra la proporción áurea (4,62) y la armonía entre el FM y los diámetros anteroposterior y transversal del cráneo. El objetivo de este estudio fue examinar la existencia de esta relación en las imágenes de TC e investigar si esta relación cambia según los tipos de forma del FM. Se examinaron 402 imágenes de TC de adultos pertenecientes a la población turca. Se midieron la longitud craneal máxima, el ancho craneal máximo y la longitud del FM y la anchura del FM. Las diferentes formas del FM se clasificaron macroscópicamente. Se registró el número e incidencia de cada tipo en el cráneo estudiado. En las 402 imágenes de TC utilizadas en el estudio, se detectaron 12 formas de FM. Tres de los tipos de formas observados en nuestro estudio no se han reportado en la literatura consultada describiéndose por primera vez. Además, nuestros datos respaldaron que al usar la proporción de 4,62, había una armonía importante entre el cráneo y el FM. Los resultados obtenidos de la investigación muestran que existe una relación de 4,62 entre las medidas craneales básicas y las dimensiones de la FM en todas las formas excepto en la triangular. Se cree que la repetición de los coeficientes calculados en más formas de FM diferentes contribuirá a la eficacia de la proporción áurea propuesta.

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INTRODUCTION: The macroscopic dynamic of fetal penis development presents a pattern resembling the unfolding of a spiral, so congenital ventral penile curvature could indicate that this natural sequence has been interrupted. Our aim in this article is to offer a mathematical model of congenital ventral curvature of the penis. MATERIALS AND METHODS: Five individuals who presented with congenital ventral penile curvature and three who presented with acquired penile ventral curvature due to Peyronie's disease were evaluated. The penises were photographed during an induced erection test and the penile curvature patterns were compared with an equiangular spiral. When an association was found, a potential relationship to the golden spiral-a type of equiangular spiral-was also assessed. The mathematical spiral relationships were analyzed using Wolfram CDF Player® (Logarithmic Spiral) and PhiMatrix® software. The Wolfram software generated logarithmic spirals equivalent to the penile curvature with appropriate mathematical values. The PhiMatrix software, which builds any golden spirals from golden rectangles, was used to check whether the spiral was golden as well as equiangular. RESULTS: An equiangular spiral that was also golden was found in all cases of congenital ventral penile curvature. In contrast, none of the acquired penile ventral curvature cases showed a specific pattern. CONCLUSION: Congenital ventral penile curvature has the mathematical pattern of a golden spiral. Our results offer a mathematical algorithm for potential use in surgical reconstruction procedures, regenerative medicine, tissue engineering, robotics, and body-machine interfaces.

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According to Giorgio Vasari (1511-1574), the great genius of anatomy, Michelangelo Buonarroti (1475-1564), in painting the frescoes on the ceiling of the Sistine Chapel (1508-1512), demonstrated to the world a new dimension/perspective of painting, especially in the sublime manner with which the artist represented the anatomical details of the characters that made up his frescoes. Since then, the Sistine Chapel has received millions of tourists annually, who marvel at the anatomical beauty of the characters depicted on its ceiling. It has also received many scholars of art and even anatomists, who have often tried to infer theses and explanations regarding Michelangelo's real intentions in elaborating this great work. However, even after five centuries, the Vatican's own official explanations of the true intentions of the artist remain quite uncertain. In an attempt to elucidate Michelangelo's possible intentions in the design of this memorable work, this article presents unpublished evidence that all the frescoes on the ceiling of the Sistine Chapel were organized by the artist according to a code based on the number of characters in each fresco, the gematria of the Hebrew/Greek alphabet, and the Golden Ratio. This decoding process could reveal a key factor influencing the artist's intentions in ordering the elements in this work. Clin. Anat. 31:948-955, 2018. © 2018 Wiley Periodicals, Inc.

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The aim of the present study is to determine the frequency of the occurrence of supraorbital foramen/notch (SOF/N) in the skulls of the people who lived in the modern era and the late Byzantine era, to determine the symmetry and the asymmetry between the two halves of the skulls by measuring the linear distance to various landmarks, to check the consistency between the location of the SOF/N and the golden ratio by calculating the ratio between linear distances and to evaluate the differences between the skulls from both historical periods. In the study, the frequency of the occurrence of the supraorbital notch in the skulls from the Byzantine era was found to be 26.60 % on the right and 13.30 % on the left while it was 14.30 % on the right and 9.52 % on the left in the skulls belonging to modern humans. In the skulls belonging to the Byzantine era, the average distance between SOF/N and the sagittal axis passing along the lateral orbital wall was found to be 34.81±2.51 mm and 32.99±2.81 mm respectively on the right and the left while it was 33.14±2.19 mm and 33.39±2.06 mm in the skulls belonging to modern era. The average distance between the SOF/N and the sagittal plane passing along the midline of the skull was found to be 24.55±2.79 mm and 21.57±2.44 mm on the right and the left respectively in the skulls belonging to the Byzantine era while it was 0.04±3.30 mm and 20.96±2.37 mm in the skulls belonging to the modern era. The average distance between the SOF/N and sagittal plane passing along the medial orbital wall was found to be 23.78±3.60 mm and 23.81±3.20 mm on the right and the left respectively in the skulls belonging to the Byzantine era while it was found to be 22.23±3.29 mm and 23.97±1.93 mm in the skulls belonging to the modern era. The average value of the distance between the sagittal planes passing along the lateral and medial sides of the orbit and the ratios between the distance from SOF/N to the sagittal plane passing along the lateral side of the orbit was found to be 1.47±0.21 mm and 1.60±0.08 mm respectively in the skulls belonging to the Byzantine era and the modern era. No significant difference was found between this ratio and the golden ratio; the average value of the ratios between the distance from the SOF/N to the sagittal plane passing along the midline of the skull and the distance from the SOF/N to the sagittal plane passing along the lateral side of the orbit was found to be 0.98±0.26 mm and 1.04±0.36 mm respectively in the skulls belonging to the Byzantine era and the modern era. A significant difference was found between this ratio and the golden ratio for both historical periods (modern society and late Byzantine period) (p <0.005). The comparison of the relevant anatomic characteristics of the SOF/N is very important for anthropologists while a broad knowledge on proportional calculations regarding morphometric values and the location are important for reconstructive surgeons and the experts in forensics and pain control.

El objetivo fue determinar la frecuencia del foramen y incisura supraorbitaria (FSO/E) en cráneos de individuos de la era moderna y aquellos de la era bizantina, para determinar la simetría y asimetría entre las dos mitades de cráneos, mediante la medición de la distancia lineal a varios puntos de referencia, para comprobar la coherencia entre la ubicación del FSO/ y la proporción áurea, a través del cálculo de la relación entre las distancias lineales. Además, se evaluaron las diferencias entre los cráneos de periodos históricos. En el estudio, se determinó que la frecuencia registrada de la incisura supraorbitaria en los cráneos de la época bizantina fue del 26,60% en el lado derecho y 13,30 % en el izquierdo; mientras que en cráneos humanos modernos fue de 14,30% en el lado derecho y 9,52 % en el izquierdo. En los cráneos pertenecientes a la época bizantina, la distancia media entre FSO/E y el eje sagital a lo largo de la pared lateral de la órbita fue de 34,81±2,51 mm y 32,99±2,81 mm, a la derecha e izquierda, respectivamente; mientras que en cráneos de la era moderna fue de 33,14±2,19 mm y 33,39±2,06 mm, a la derecha e izquierda, respectivamente. La distancia media entre la FSO/E y el plano sagital que pasa a lo largo de la línea mediana del cráneo, en los cráneos de la época bizantina, fue de 24,55±2,79 mm y 21,57±2,44 mm a la derecha e izquierda, respectivamente; mientras que fue de 0,04±3,30 mm y 20,96±2,37 mm en los cráneos de la era moderna. La distancia media entre la FSO/E y el plano sagital que pasa a lo largo de la pared medial de la órbita, en los cráneos pertenecientes a la época bizantina fue de 23,78±3,60 mm y 23,81±3,20 mm, a la derecha e izquierda, respectivamente; mientras que en los cráneos pertenecientes a la era moderna fue de 22,23±3,29 mm y 23,97±1,93 mm, a la derecha e izquierda, respectivamente. Se determinó que el valor medio de la distancia entre los planos sagitales a lo largo de los márgenes lateral y medial de la órbita y las proporciones entre la distancia desde FSO/E al plano sagital a lo largo del lado lateral de la órbita fueron de 1,47±0,21 mm y 1,60±0,08, respectivamente, en los cráneos pertenecientes a la época bizantina y la era moderna. No encontramos diferencias significativas entre esta relación y la proporción áurea. El valor medio de las relaciones entre la distancia de la FSO/E al plano a lo largo de la línea mediana del cráneo y la distancia de la FSO/E al plano sagital a lo largo de la pared lateral de la órbita fueron de 0,98±0,26 mm y 1,04±0,36 mm, respectivamente, en los cráneos pertenecientes a la era bizantina y la era moderna. No se encontró una diferencia significativa entre esta relación y la proporción áurea en ambos períodos históricos (la sociedad moderna y el periodo bizantino tardío) (p <0,005). La comparación de características anatómicas relevantes para el FSO/E son importantes para los antropólogos, donde un conocimiento integro de cálculos proporcionales con respecto a los valores morfométricos, mientras que para los cirujanos reconstructivos y los expertos en medicina forense es importante su ubicación.

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The aim of this study was to examine the morphometric characteristics of the infraorbital foramen of skulls of people living in modern society and in the late Byzantine period, to ascertain the symmetry or asymmetry of the two halves of the skulls by measuring the linear distance between various landmarks, to evaluate at the conformity between the infraorbital foramen and the golden ratio by calculating the ratios between these linear distances, and to set out the differences or similarities between the skulls of these different periods. It was found in the study that the morphometric characteristics of the infraorbital foramen in skulls of the modern period were 47.05 % circular, 41.17 % oval and 11.76 % atypical (semilunar and triangular) on the right, and 70.58 % circular and 29.41 % oval on the left, while those of the Byzantine period were 46.06 % circular and 53.3% oval on the right, and 50% circular and 50 % oval on the left. It was found that the measurements across the infraorbital foramen of the Byzantine skulls averaged 2.93±1.05 mm and 3.15±1.03 mm on the right side and 2.62±0.97 mm and 3.16±0.68 mm on the left vertically and horizontally respectively, and those of the modern period measured 2.32±0.50 mm and 3.00±0.92 mm on the right and 2.48±0.45 mm and 2.76±0.65 mm on the left vertically and horizontally respectively. The ratio between the distances from the outer orbital wall of the IOF to the sagittal plane and the piriform aperture were 1.46±0.25 mm and 1.40±0.21 mm for left and right respectively for the Byzantine skulls, and 1.24±0.24 mm and 1.29±0.42 mm for the modern skulls. A significant difference was found between the ratio obtained and the golden ratio for each of the periods (modern and Byzantine) (p<0.005). A comparison of anatomical characteristics of the infraorbital foramen of people living at different historical periods is important for anthropologists and knowing morphological types and amassing knowledge on the proportional calculation of location is important for dentistry, maxillofacial surgery and algology.

El objetivo fue examinar las características morfométricas de los forámenes infraorbitarios en cráneos de personas que viven en la sociedad moderna y aquellas que vivieron en el periodo bizantino tardío, para determinar la simetría o asimetría de las dos mitades de los cráneos. Se realizó la medición de la distancia lineal entre varios puntos de referencia, para evaluar la conformidad entre el foramen infraorbitario (FO) y la proporción áurea, mediante el cálculo de la relación entre esas distancias lineales, así como las diferencias y similitudes entre los cráneos de estos períodos. En relación a las características morfométricas, se encontraron en cráneos de la época moderna FO circulares (47,05%), ovalados (41,17 %) y atípicos (semilunar y triangular) en un 11,76 %, del lado derecha, y en el lado izquierdo estos porcentajes correspondieron a circulares (70,58 %) y ovalados (29,41%). Por su parte, los cráneos del período bizantino presentaron FO circulares (46,06 %) y ovalados (53,3 %) del lado derecho, mientras que del lado izquierdo fueron circulares (50 %) y ovalados (50 %). Se encontró que las medidas del FO de los cráneos bizantino tuvo un promedio de 2,93±1,05 mm y 3,15±1,03 mm en el lado derecho, y 2,62±0,97 mm y 3,16±0,68 mm a la izquierda, vertical y horizontal, respectivamente; y en el período moderno midieron 2,32±0,50 mm y 3,00±0,92 mm a la derecha y 2,48±0,45 mm y 2,76±0,65 mm a la izquierda, en medidas verticales y horizontales respectivamente. La relación entre las distancias de la pared orbitaria exterior del FO al plano sagital y la apertura piriforme fue de 1,46±0,25 mm y 1,40±0,21 mm, a izquierda y derecha respectivamente, para los cráneos bizantinos; y 1,24±0,24 mm y 1,29±0,42 mm para los cráneos modernos. Se encontró una diferencia significativa entre la proporción obtenida y la proporción áurea para cada uno de los períodos (moderno y bizantino) (p <0,005). La comparación de las características anatómicas del FO de individuos que vivieron en diferentes períodos históricos es valioso para los antropólogos y los conocimientos sobre el cálculo proporcional de la ubicación del FO son importantes en la odontología, cirugía maxilofacial y el estudio del dolor.

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Art and anatomy were particularly closely intertwined during the Renaissance period and numerous painters and sculptors expressed themselves in both fields. Among them was Michelangelo Buonarroti (1475-1564), who is renowned for having produced some of the most famous of all works of art, the frescoes on the ceiling and on the wall behind the altar of the Sistine Chapel in Rome. Recently, a unique association was discovered between one of Michelangelo's most celebrated works (The Creation of Adam fresco) and the Divine Proportion/Golden Ratio (GR) (1.6). The GR can be found not only in natural phenomena but also in a variety of human-made objects and works of art. Here, using Image-Pro Plus 6.0 software, we present mathematical evidence that Michelangelo also used the GR when he painted Saint Bartholomew in the fresco of The Last Judgment, which is on the wall behind the altar. This discovery will add a new dimension to understanding the great works of Michelangelo Buonarroti.

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Michelangelo Buonarroti (1475-1564) was a master anatomist as well as an artistic genius. He dissected numerous cadavers and developed a profound understanding of human anatomy. Among his best-known artworks are the frescoes painted on the ceiling of the Sistine Chapel (1508-1512), in Rome. Currently, there is some debate over whether the frescoes merely represent the teachings of the Catholic Church at the time or if there are other meanings hidden in the images. In addition, there is speculation regarding the image of the brain embedded in the fresco known as "The Creation of Adam," which contains anatomic features of the midsagittal and lateral surfaces of the brain. Within this context, we report our use of Image Pro Plus Software 6.0 to demonstrate mathematical evidence that Michelangelo painted "The Creation of Adam" using the Divine Proportion/Golden Ratio (GR) (1.6). The GR is classically associated with greater structural efficiency and is found in biological structures and works of art by renowned artists. Thus, according to the evidence shown in this article, we can suppose that the beauty and harmony recognized in all Michelangelo's works may not be based solely on his knowledge of human anatomical proportions, but that the artist also probably knew anatomical structures that conform to the GR display greater structural efficiency. It is hoped that this report will at least stimulate further scientific and scholarly contributions to this fascinating topic, as the study of these works of art is essential for the knowledge of the history of Anatomy.

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In rodents, the novel object preference test has been used as a behavioral parameter for evaluation of neotic exploratory behavior, and also for memory consolidation tasks. Geometric patterns of this preference are poorly understood, and may vary among species. We evaluated in Wistar rats (Rattus norvergicus) a possible exploration preference considering aluminum tripartite rounded and cylindrical objects of different proportions: 1.2; 1.618; 1.8. At the first day, animals were exposed to 1.2; 1.6 and 1.8 rounded objects. At 24h after, these animals were exposed to the same objects, together with three new steel cylindrical objects (same proportions). ANOVA and T tests were used to quantify object exploration for each animal (p<0.05). Data analysis pointed to a longer exploration time of the object 1.2 at the three different protocols indicating a preference pattern on the first day exposition. On the second day the exploration was similar in both familiar and unfamiliar objects, revealing no novel object preference for cylinders. However, we found an object preference related to the 1.2 proportion (balls plus cylinders), in two of three position protocols. In addition, on a single exposition with both cylinders and rounded objects, rats revealed a rounded object preference. The 1.2 preference disclosed by rats also reflected the proportion of their body. From nine main measures of body ratios, seven were close to 1.2 ratio. The correspondence between body ratios and object preference may be explained by habituation learning and by sexual selection, and highlight innate factors regarding aesthetic preferences among species.

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